Temperature and Average Kinetic Energy

[jmsmith]

In school I was always taught that temperature is proportional to the average kinetic energy of the individual particles in that object. However, I recently encountered a brief statement in a 1999 issue of The Physics Teacher that this is only really the case for ideal gases. Here is the quote from PT:

"Quantum effects can lock out certain energy levels, thus prohibiting energy exchanges between particles. For instance, at room temperature, the average kinetic energy of the electrons in a metal is of the order of several eV or tens of thousands K." -PT 1999 Survey of High School Physics Texts (authored by the PT editorial staff)

How correct is this? How wrong is it to generally say that temperature is proportional to the average kinetic energy of the individual particles? What would be a good way to think about/define temperature?

Also, if metals have all that KE, why don't they feel incredibly hot? Why doesn't energy tend to flow out of a metal rapidly and into the metal's less energetic surroundings? Why doesn't the metal either A) radiate this energy away via electromagnetic waves, or B) pass this energy to surroundings/objects that are in thermal contact with the metal?

 

[murrdpirate0]

I'm pretty sure that what we feel as "hotness" and define as temperature only depends on the motions of the molecules, not the electrons within them. Which makes sense because if you picture a bunch of molecules hitting your hand at high velocities, the only thing that matters to you is how fast the molecules are going, not how fast the electrons are orbiting the atoms.

I'm also pretty sure that temperature is always proportional to kinetic energy. The constant of proportionality may change for non-ideal gases (and may vary with temperature for these gases) due to inter-molecular forces, but it will still be dependent on kinetic energy.

 

[Count Iblis]

It is completely wrong to say that temperature is the average kinetic energy. Of course, it follows from the equipartition theorem of classical statistical mechanics that the average kinetic energy is proportional to temperature for a classical system. But you cannot use that to define temperature.

Temperature is a purely statistical concept. It is defined as:

1/(k T) = d[Log(Omega(E))]/dE

where Omega(E) is the number energy eigenstates in a small interval around the energy E of the system.

 

[jmsmith]

I'm pretty sure that what we feel as "hotness" and define as temperature only depends on the motions of the molecules, not the electrons within them. Which makes sense because if you picture a bunch of molecules hitting your hand at high velocities, the only thing that matters to you is how fast the molecules are going, not how fast the electrons are orbiting the atoms.

I'm also pretty sure that temperature is always proportional to kinetic energy. The constant of proportionality may change for non-ideal gases (and may vary with temperature for these gases) due to inter-molecular forces, but it will still be dependent on kinetic energy.

But if temperature is proportional to kinetic energy, then what about the kinetic energy of those electrons? What about conductors (where electrons do not really "orbit" atoms)? Also, what prevents the kinetic energy of the electrons in a conductor from passing to my hands from that conductor?

 

[jmsmith]

It is completely wrong to say that temperature is the average kinetic energy. Of course, it follows from the equipartition theorem of classical statistical mechanics that the average kinetic energy is proportional to temperature for a classical system. But you cannot use that to define temperature.

Temperature is a purely statistical concept. It is defined as:

1/(k T) = d[Log(Omega(E))]/dE

where Omega(E) is the number energy eigenstates in a small interval around the energy E of the system.

One answer inspires many more questions. I would greatly appreciate any good answers, but understand that people are busy:

• I re-work the equation to basically mean that temperature is proportional to the rate of change in energy with entropy. How correct is this?
• How does this equation connect with the fact that thermal energy tends to flow from objects at high temperatures to objects at lower temperatures?
• When does a system typically become non-classical? Under what conditions would I expect to find a non-classical system?
• What is the connection between the equation above and the classical idea that temperature is proportional to average kinetic energy? (or that increase in temperature is proportional to the amount of energy added to the system, assuming the system is not undergoing a phase change?)
• Finally, how would a mechanical engineer think of temperature? How would a chemist/chemical engineer think of temperature? How different would these be from the way that a physicist thinks of temperature?